Relative regular modules. Applications to von Neumann regular rings

TL;DR

This paper introduces the concept of relative regular modules to transfer regularity properties between categories and rings, providing new proofs and insights into von Neumann regular rings and their Morita invariance.

Contribution

It develops the notion of regular objects relative to others in categories, leading to new proofs of von Neumann regularity invariance and analysis of Morita rings.

Findings

Von Neumann regularity transfers via excellent extensions

Morita invariance of von Neumann regularity proved using relative modules

Regularity of Morita rings analyzed

Abstract

We use the concept of a regular object with respect to another object in an arbitrary category, defined in \cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod, when $S$ is an excellent extension of the ring $R$. Consequently, we obtain a result of \cite{ps}: if $S$ is an excellent extension of the ring $R$, then $S$ is von Neumann regular ring if and only if $R$ is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular property of a ring is Morita invariant. Finally, the von Neumann regularity of the Morita ring is investigated.